Chelsea, themselves transformed a decade ago by a cascade of cash from a then unknown overseas investor, had labelled their Champions League opponents as "dark horses" on their own website in the buildup to the quarter-final first leg. Yet that would only have been accurate if the black steed in question was wearing a bright pink feather boa and leaving a trail of high denomination Euros in its wake.

Qatar's trolley dash around world sport could not have been more eye catching, even without the controversy and turmoil that has surrounded its successful World Cup 2022 bid.

The PSG purchase was just one of a number that have been marked down as an attempt to both secure the Gulf state's future beyond the point when its natural gas and oil reserves run dry, and afford it international profile.

That process reached its natural conclusion with the vote in December 2010 to award the 2022 World Cup to Qatar, a decision that remains mired in controversy as Fifa and organisers grapple with the desperate plight of the 1.4m migrant workers and questions still swirling around the bidding process.

Yet it goes much broader and deeper, also encompassing a hitherto largely ignored parallel network of sports TV channels, branded BeIn Sports, that also play a significant strategic role. Fans in France watched the Champions League game live on Canal Plus, but the French broadcaster now shares rights to the competition with its Qatar-owned rival.

From the shirts of fellow quarter-finalists Barcelona emblazoned with Qatar Airways to the Bayern Munich side that regularly troop to Doha for their winter training breaks, and the network of Aspire Academies that criss-cross the globe, the emirate is transforming itself into a sporting superpower.

Most noticeably, the trajectory blazed by PSG since the club was bought in April 2011 by Qatar Sports Investments, an arm of the $115bn Qatar Investment Authority that has stakes in a global portfolio that stretches from Credit Suisse to the owner of Heathrow airport, is even steeper than those other grand projects at Stamford Bridge and Manchester City's Etihad Stadium.

The official reason why PSG were seized upon by QSI and their president, Nasser al-Khelaifi, was their huge potential in what is the biggest one-club city in Europe. Since then, partly through acquisitions including Zlatan Ibrahimovic, Edinson Cavani and David Beckham, the club has been successfully rebranded in the eyes of the world and – perhaps as importantly – in the eyes of Parisians. A once down‑at-heel, down-on-its luck club playing in front of two factions of home fans who used to fight one another after each game has been transformed into an upmarket lifestyle choice. "We have a very clear vision, to be honest," Khelaifi told the Financial Times last week. "In five years, we want to be one of the best clubs in Europe and to win the Champions League. And our brand to be worth €1bn. And we will be there."

The ambition is to become one of the top 10 "franchises" in any sport on the planet and they are unafraid to spend big to do it. PSG expect annual revenues to top €500m this year but a large chunk of that will come from a much-questioned €200m-a-year sponsorship deal with the Qatar Tourism Authority, backdated to 2012. The unprecedented deal will stretch the fine print of Uefa's financial fair play rules to their limit.

L'Equipe was the latest publication to wonder last week how the club, which has already been heavily subsidised by its owners, will cope with the FFP restrictions. Rivals are queuing up to see how tough Uefa will be.

The unofficial reason that Qatar alighted upon PSG lies in the already close links between the two countries and a much-discussed lunch between the then French president, Nicolas Sarkozy, Uefa's president Michel Platini and Tamim bin Hamad al-Thani, who has since succeeded his father, Hamad, as the Qatar's emir.

Platini has admitted that Sarkozy wanted the Qataris to buy PSG and secure his vote – and with it his influence in the looming World Cup vote over other members of the European bloc – but denies he was influenced by him. The Uefa president's son, Laurent, was subsequently offered the job as chief executive of Burrda, the QSI-owned sports clothing company, which Platini Snr says is completely unconnected to him voting for Qatar.

Sarkozy later became a passionate advocate of the Qatar World Cup and was one of the first to lobby for the football calendar to be changed to allow it to take place in winter rather than the searing summer heat.

The high-profile Qatari spending spree has been well-documented. But the parallel attempt to invest hundreds of millions of pounds in a global network of TV channels is also key to its ambitions. The same man, the suave Khelaifi, oversees both the PSG project and the global suite of sports channels.

As with PSG, it is in France where its impact has been felt most keenly, sparking a bitter battle with the incumbent Canal Plus that has parallels with the fierce bidding war for top-flight football between Sky and BT in the UK.

Arriving on the scene in France in 2012, a year after the PSG acquisition, the rebranded al-Jazeera Sport surprised Canal Plus with a clever bidding strategy. It left it with eight of the 10 weekly Ligue 1 games – albeit not the first-choice matches – for a third of the price paid by its rival. Between them, the two broadcasters pay a total of €607m over the course of the three-year deal. BeIn Sports subsequently landed Champions League rights, prompting a competition complaint to regulators from Canal Plus, the establishment player that believed the brash newcomer was being run at an uncompetitive loss. The BeIn Sports channels massively undercut the existing player but it insists it has long‑term ambitions to become profitable – the rhetoric is not unlike that surrounding PSG.

To take advantage of the rivalry, the Ligue de Football Professionnel, the body that runs the French league, has brought forward the auction for its 2016 to 2020 rights – prompting another complaint to regulators from Canal Plus.

"The football league sold us rights for four years, until June 2016. We want to benefit from it serenely. I do not understand this precipitation," said the Canal Plus chief executive, Bertrand Meheut. Three days after the LFP auction, Uefa will sell the rights to the Champions League for 2015 to 2018 in France.

If the two auctions go BeIn Sports' way, it would strengthen Qatar's grip on French sport, as owner of the biggest club in the country and of the most influential sports broadcaster.

When Sky attempted a similar strategy in England, launching a bid for Manchester United, it was stymied by a fan backlash and competition regulators. In France, they have been largely ushered in with open arms.

BeIn Sports also launched aggressively into the US, using the rights for the Spanish and Italian leagues to put it on the map. In Canada, it has jumped through complex regulatory hoops to acquire a Canadian subsidiary and satisfy licence conditions.

It has also launched operations in Hong Kong and Indonesia. During the last Premier League rights auction it thought carefully about entering the £3.1bn battle between Sky and BT before concluding there was better value, and potentially more leverage, to be found elsewhere.

That could yet change, with the Premier League also considering launching an early auction to take advantage of the likely competition.

Rumours persist that BeIn Sports could take a substantial stake in Silvio Berlusconi's Mediaset, which would give it a foothold not only in Italy but also in Spain, where Mediaset owns 41% of the broadcaster that has the rights to the games played by the national side.

Sports broadcasting insiders believe that its ambitions are limitless and that it is engaged in a race with Fox International and, to a lesser extent, Discovery, to establish a network of sports channels around the world to rival ESPN and Star Sports.

"They are voracious with their ambition and they have all the money in the world to match it. There is nothing they can't do," said one. "What drives it is their need to be one of the top five broadcasters in the world by 2022."

It has not all been plain sailing. Some Italian clubs are now wondering whether BeIn Sports gives them enough profile in the US, seen as a key growth market. Others wonder what would happen if the Emir, who is said to be close to Khelaifi, decided that its sporting investments were having the opposite of the desired effect and doing more harm than good to the country's global image.

Such a scenario is considered unlikely but, if it has to carry on subsidising large losses, the sports arm does not have the protection that al-Jazeera's news network would probably enjoy given its use as a diplomatic tool. In the meantime, on the pitch and in its studios, the Qatari incursion into global sport will continue.

Sitting by his pool in Doha, Richard Keys – the former Sky Sports anchor who resigned amid a series of sexism allegations – considers the ambition of the company that hired him to front its Premier League coverage in its native Middle Eastern markets.

"When I came over here, I realised there were other broadcasters in the world. They showed el clásico in America, in France, in Spain, here and in Indonesia. That suggests to me that what they want to do is what [Rupert] Murdoch has always wanted to do, and become the global broadcaster."

Events in France over the coming weeks, assuming the LFP auction takes place, and in Switzerland, where Uefa's lawyers and accountants are scrutinising PSG's books, will help shape the immediate future of an unprecedented global experiment in sport and television that could have as yet unappreciated ramifications for both.